Finite Group Databasebeta

← #28498588

Subgroups — #28498588

Order 15104.

306 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433074C211yesmaximal, characteristic
2#21425897C211yesmaximal, characteristic, complemented
3#21425850C211yesmaximal, characteristic, complemented
4#21422079C211yesmaximal, characteristic, complemented
5#21422077C211yesmaximal, characteristic, complemented
6#21422054C211yesmaximal, characteristic, complemented
7C59×#1330C211yesmaximal, characteristic, complemented
8#16837934C411yescharacteristic
9C2×#14273666—21no—
10#16837532C2211yescharacteristic
11#16837526C411yescharacteristic
12#16837471—21no—
13D4×C59⋊C8C411yescharacteristic
14(C22⋊D59)⋊C8—21no—
15#16837460—21no—
16#16837458C411yescharacteristic
17#16837360—21no—
18C4×C8×D59C2211yescharacteristic
19C4⋊(C59⋊C16)C2211yescharacteristic
20Dic59⋊C16—21no—
21Dic59⋊C16—21no—
22(C2×D59)⋊C16—21no—
23#16837178C2211yescharacteristic
24C59×#436C2211yescharacteristic
25C59×C4⋊C16C2211yescharacteristic
26C59×#346C2211yescharacteristic
27C22×C59⋊C8—21no—
28C2×C8×D59D411yescharacteristic
29C2×C8×D59D411yescharacteristic
30#14273720—21no—
31#14273718C811yescharacteristic
32#14273714C811yescharacteristic
33#14273709—21no—
34#14273707C811yescharacteristic
35#14273698C811yescharacteristic
36C4×Dic118C2×C411yescharacteristic
37C4×C22⋊D59—21no—
38#14273686—21no—
39C42×D59C2×C411yescharacteristic
40#14273666—42no—
41C2×C59⋊C16—21no—
42C2×C59⋊C16—21no—
43C4⋊(C59⋊C8)C2×C411yescharacteristic
44Dic59⋊C8—21no—
45Dic59⋊C8—21no—
46C22⋊(C59⋊C8)—21no—
47(C2×D59)⋊C8—21no—
48#14273637—21no—
49#14273637—21no—
50C4×C59⋊C8C2×C411yescharacteristic
51C8×Dic59—21no—
52C8×Dic59D411yescharacteristic
53C8×Dic59D411yescharacteristic
54C2×M16×C59—21no—
55C4×D4×C59C2×C411yescharacteristic
56C2×C16×C59—21no—
57C2×C16×C59—21no—
58C4⋊C8×C59C2×C411yescharacteristic
59C22⋊C8×C59—21no—
60C4×C8×C59C2311yescharacteristic
61C2×D4×C59C2×C811yescharacteristic
62C22×C4×C59—21no—
63C2×C22⋊D59—21no—
64C2×C4×D59C22⋊C411yescharacteristic
65C2×C4×D59C2×C811yescharacteristic
66C2×C4×D59M1611yescharacteristic
67C2×Dic118C2×C811yescharacteristic
68M16×C59—41no—
69M16×C59—41no—
70C2×C8×C59—21no—
71C2×C8×C59C2×D411yescharacteristic
72C2×C8×C59C2×D411yescharacteristic
73C4⋊C4×C59C2×C811yescharacteristic
74C22⋊C4×C59—21no—
75C42×C59C22×C411yescharacteristic
76C22⋊Dic59—21no—
77C22⋊Dic59—21no—
78C22⋊Dic59—21no—
79(C2×D59)⋊C4—21no—
80C4⋊Dic59—21no—
81C4⋊Dic59—21no—
82C4⋊Dic59C2×C811yescharacteristic
83Dic59⋊C4—21no—
84Dic59⋊C4—21no—
85C4×Dic59—21no—
86C4×Dic59C22⋊C411yescharacteristic
87C4×Dic59C2×C811yescharacteristic
88C4×Dic59M1611yescharacteristic
89C2×C59⋊C8—21no—
90C2×C59⋊C8—42no—
91C2×C59⋊C8C22⋊C411yescharacteristic
92C2×C59⋊C8C22⋊C411yescharacteristic
93C8×D59—21no—
94C8×D59—21no—
95C8×D59—41no—
96C16×C59—41no—
97C16×C59—41no—
98C59⋊C16—41no—
99C59⋊C16—41no—
100C23×C59—21no—
101C22×D59C22⋊C811yescharacteristic
102D4×C59—41no—
103C2×C4×C59—21no—
104C2×C4×C59—41no—
105C2×C4×C59—41no—
106C2×C4×C59C2×C22⋊C411yescharacteristic
107C2×C4×C59C22×C811yescharacteristic
108C2×C4×C59C2×M1611yescharacteristic
109C22⋊D59—42no—
110C2×Dic59—21no—
111C2×Dic59—21no—
112C2×Dic59C22⋊C811yescharacteristic
113C2×Dic59C22⋊C811yescharacteristic
114C2×Dic59C22⋊C811yescharacteristic
115C4×D59—21no—
116C4×D59—21no—
117C4×D59—41no—
118C4×D59—41no—
119Dic118—41no—
120C8×C59—21no—
121C8×C59—41no—
122C8×C59AGL(1,ℤ/8)11yescharacteristic
123C8×C59#13711yescharacteristic
124C59⋊C8—21no—
125C59⋊C8—21no—
126C59⋊C8—41no—
127#11627—591nomaximal, Sylow
128C22×C59—41no—
129C22×C59—41no—
130C22×C59—41no—
131C22×C59C2×C22⋊C811yescharacteristic
132C2×D59—21no—
133C2×D59—21no—
134C2×D59—21no—
135C4×C59—21no—
136C4×C59—21no—
137C4×C59—41no—
138C4×C59#41711yescharacteristic
139C4×C59#42111yescharacteristic
140Dic59—21no—
141Dic59—21no—
142Dic59—21no—
143Dic59—41no—
144Dic59—41no—
145#2966—591no—
146#2163—591no—
147#2150—591no—
148#1338—591no—
149#1338—591no—
150#1330—591no—
151#1330D5911yescharacteristic, complemented
152C2×C59—81no—
153C2×C59#147711yescharacteristic
154C2×C59#211511yescharacteristic
155C2×C59#211711yescharacteristic
156D59—41no—
157#529—591no—
158C2×M32—1181no—
159#446—591no—
160C8×Q8—591no—
161#436C2×D5911yescharacteristic
162#435—1181no—
163C8×D4—591no—
164C8×D4—1181no—
165#433—1181no—
166#432—591no—
167#405—1181no—
168C2×C4×C8—591no—
169C4⋊C16—591no—
170C4⋊C16—1181no—
171C4⋊C16—1181no—
172C4⋊C16C2×D5911yescharacteristic
173C22⋊C16—1181no—
174#346—591no—
175#346C2×D5911yescharacteristic
176C59#1162711yescharacteristic, complemented, Sylow
177C2×M16—21no—
178C22×C8—591no—
179C22×C8—591no—
180C22×C8—1181no—
181#126—1181no—
182#125—591no—
183#124—591no—
184#123—1181no—
185C22⋊Q8—591no—
186#121—591no—
187C4×Q8—591no—
188C4×D4—1181no—
189C4×D4C4×D5911yescharacteristic
190#117—1181no—
191C2×C42—591no—
192M32—2362no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—1181no—
196C2×C16—1181no—
197C4⋊C8—591no—
198C4⋊C8—1181no—
199C4⋊C8—1181no—
200C4⋊C8C4×D5911yescharacteristic
201C22⋊C8—21no—
202C22⋊C8—1181no—
203C22⋊C8—1181no—
204#97—1181no—
205#97—1181no—
206C4×C8—591no—
207C4×C8—591no—
208C4×C8—591no—
209C4×C8—1181no—
210C4×C8C22×D5911yescharacteristic
211C2×Q8—591no—
212C2×D4—1181no—
213C2×D4C8×D5911yescharacteristic
214C22×C4—21no—
215C22×C4—591no—
216C22×C4—591no—
217C22×C4—591no—
218M16—41no—
219M16—41no—
220C2×C8—21no—
221C2×C8—591no—
222C2×C8—591no—
223C2×C8—1181no—
224C2×C8—1181no—
225C2×C8—1181no—
226C2×C8—2361no—
227C2×C8—2362no—
228C2×C8D4×D5911yescharacteristic
229C2×C8D4×D5911yescharacteristic
230C4⋊C4—591no—
231C4⋊C4—1181no—
232C4⋊C4—1181no—
233C4⋊C4—1181no—
234C4⋊C4—1181no—
235C4⋊C4C8×D5911yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—1181no—
238C22⋊C4—1181no—
239C22⋊C4—1181no—
240C22⋊C4—1181no—
241C42—591no—
242C42—591no—
243C42—591no—
244C42—1181no—
245C42C2×C4×D5911yescharacteristic
246C16—41no—
247C16—41no—
248C16—2361no—
249C16—2361no—
250C23—21no—
251C23—591no—
252Q8—2361no—
253D4—41no—
254D4—2362no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—591no—
259C2×C4—591no—
260C2×C4—591no—
261C2×C4—1181no—
262C2×C4—1181no—
263C2×C4—1181no—
264C2×C4—1181no—
265C2×C4—2361no—
266C2×C4—2361no—
267C2×C4C22⋊C4×D5911yescharacteristic
268C2×C4C2×C8×D5911yescharacteristic
269C2×C4M16×D5911yescharacteristic
270C8—21no—
271C8—41no—
272C8—1181no—
273C8—1181no—
274C8—2361no—
275C8#1427375111yescharacteristic
276C8#1427375611yescharacteristic
277C22—41no—
278C22—41no—
279C22—41no—
280C22—1181no—
281C22—1181no—
282C22—1181no—
283C22C22⋊C8×D5911yescharacteristic
284C4—21no—
285C4—21no—
286C4—41no—
287C4—1181no—
288C4—1181no—
289C4—1181no—
290C4—2361no—
291C4—2361no—
292C4#1683740011yescharacteristic
293C4#1683741711yescharacteristic
294C2—81no—
295C2—2361no—
296C2#2142248011yescharacteristic
297C2#2142571511yescharacteristic
298C2#2142572211yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D59
commutatorC2×C4×C59C22×C8
FrattiniC4×C8C22×D59
FittingC59×#1330C2
socleC22×C59C2×C22⋊C8
prime radical#14273698C8
cosocleC4×C8×C59C23
hypercenter#1330D59
hypocenterC59#11627
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1330

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21425897C2semidirect
3. #21425850C2semidirect
4. #21422079C2semidirect
5. #21422077C2semidirect
6. #21422054C2semidirect
7. C59×#1330C2semidirect
151. #1330D59semidirect
176. C59#11627semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214330742
#214258972
#214258502
#214220792
#214220772
#214220542
C59×#13302
#1162759